Convert OBJ files to FBX files pros and cons
you have to understand that .obj files and .fbx files are following different concepts. 

The .obj file format is a simple data-format that represents 3D geometry alone. In other words, its a simple text-based file that is primary for storing Meshes.

And a .fbx file format (and collada file as an open source equivalent) can store whole scenes (Camera, Armatures, Meshes, Curves, Empties, Lights, etc).

An .obj file can store: 


Meshes information (Vertices, Edges, Faces) in multiple objects
Normals
UVs
Object information like Position, Rotation, Scale (in relation to each other!, I've often found that the obj exporter created files which had different dimensions in Blender to Meshes that had the same bounding size after importing to a different engine (Please correct me if I am wrong)
Some basic material information in a Separate .mtl File (nothing like complex cycles nodes)
Some Programs(Zbrush, XNormal, not Blender) can read and write Vertex colors to the obj file 
It is possible to create .obj animations and some engines are even capable to read them. But you have to know that blender creates for every frame a new objfiles. So, most of the time, nothing you would like to have in your realtime engine


As mentioned above, a fbx file can store a lot of things.  Essential is what your exporter(Blender) and your importer (Game engine) is capable of reading and writing. 

I definitely would recommend to use fbx or collada for any kind of exchange. I had significant problems with obj. 

to answer your questions:


  What happen with normals projections from the mesh in the conversion from file to file?



Normals should be the same in every file format



  Is it the same type of mesh (collada, mesh, etc.)?



Yes, it is the same type of mesh.



  Is it a good idea to convert obj to fbx for animations for use in an engine?



Definitely



  I create the maps, go back to Blender and see the results, but when I decided to export the whole model with textures an animation, I change the OBJ to FBX.



What you do in the process of creating your models doesn’t matter. You only want to prevent data loss. Obj is ok for texture generation with XNormal. Fbx is also ok.